Ingvar Johansson wrote:
> it makes good sense to say: "The pre-Kelvin
> temperature scales and the length scales are formally different." This
> means among other things (remember that the choice of standard unit is
> conventional), that the transformation formula between Celsius and
> Fahrenheit has the form '[°F] = 9/5 × [°C] + 32', but that between yard
> and meter has the form '[yard] = 0.91 × [meter]'. Put more generally, the
> first case fits the form 'y -> ax + b', and the latter 'y -> ax'
As long as we are talking only about formal difference, 'y -> ax' is an
instance of the form 'y -> ax + b' wherein b = 0. Ingvar is attaching
special significance to the case when b = 0, namely the "ratio" concept,
that 'a = y/x'. The formal difference is only interesting when it
becomes a criterion for classification.
> (What at a first glance may be confusing is the fact that INTERVALS of
> interval scales follow the form exemplified by ratios such as
> [interval °C] = 9/5 × [interval °F].)
>
I have yet to see a definition of "interval scale" from either David or
Ingvar. The intent seems to be that the meaning of the quantity values
is different. Is it that they don't refer to magnitudes, but rather to
"intervals"? So "interval scales" are not 'quantity value scales' (as
defined by the VIM clause that Ingvar used to correct my earlier email)? (01)
The measurement unit that is 'degree Celsius' and the measurement unit
that is 'degree Kelvin' are the same magnitude. Temperature difference
can be measured in either with the same "ratio scale". Absolute
temperature is a different concept, and it is all about how we define
the scale origin "0" with respect to points on the (putative) 'absolute
temperature axis'. And we are apparently saying that "amounts of
temperature difference" and "points on the absolute temperature axis"
are both "magnitudes" of the quantity kind "temperature". But they are
different things. Are "interval scales" scales that measure
"magnitudes" on the "absolute <quantity> axis"? (02)
This concept seems to apply to time as well. The second is defined as a
measure of difference in time (duration, elapsed time). But
International Atomic Time (TAI) identifies points on the "absolute time
axis" by their difference from a chosen 0 point. Are TAI times also
"magnitudes"? The absolute zero of time is presumably the Big Bang, but
with respect to absolute time, our current measurement technology is
more like 19th century temperature measurement. So our absolute time
scale is based on an arbitrary reference point, like Celsius choice of
the freezing temperature of water. (03)
It is easy to think of "length" as being the difference between points
in a 1-D space, and thus being parallel to 'temperature difference' and
'time difference'. But there is no 'absolute length axis' whose points
have any meaning. And it is not clear to me that mass and substance and
light intensity are differences at all. (04)
We agree that there is a difference between "interval scales" and "ratio
scales", and it can be represented mathematically by m = unit * number +
offset. But what is the definition? And how do we deal with different
kinds of magnitudes of the same quantity kind? Those are the ontology
questions. (05)
-Ed (06)
--
Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694 (07)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (08)
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